### Abstract

We present an 0(n log^{2} n) time and 0(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in 0(n log k) time, where k is the combinatorial complexity of the space of transversals and k ≤ 4n. These results find application in: (1) line-fitting between a set of n data ranges where it is desired to obtain the shortest line-of-fit, (2) finding the shortest line segment from which a convex n-vertex polygon is weakly externally visible, and (3) determining the shortest line-of-sight between two edges of a simple n-vertex polygon, for which 0(n) time algorithms are also given. AU the algorithms are based on the solution to a new fundamental geometric optimization problem that is of independent interest and should find application in different contexts as well.

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - 18th International Colloquium, Proceedings |

Publisher | Springer-Verlag |

Pages | 649-660 |

Number of pages | 12 |

ISBN (Print) | 9783540542339 |

DOIs | |

State | Published - Jan 1 1991 |

Event | 18th International Colloqulum on Automata, Languages, and Programming, ICALP 1991 - Madrid, Spain Duration: Jul 8 1991 → Jul 12 1991 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 510 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 18th International Colloqulum on Automata, Languages, and Programming, ICALP 1991 |
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Country | Spain |

City | Madrid |

Period | 7/8/91 → 7/12/91 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automata, Languages and Programming - 18th International Colloquium, Proceedings*(pp. 649-660). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 510 LNCS). Springer-Verlag. https://doi.org/10.1007/3-540-54233-7_171

**Computing shortest transversals.** / Bhattacharya, Binay; Toussaint, Godfried.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Automata, Languages and Programming - 18th International Colloquium, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 510 LNCS, Springer-Verlag, pp. 649-660, 18th International Colloqulum on Automata, Languages, and Programming, ICALP 1991, Madrid, Spain, 7/8/91. https://doi.org/10.1007/3-540-54233-7_171

}

TY - GEN

T1 - Computing shortest transversals

AU - Bhattacharya, Binay

AU - Toussaint, Godfried

PY - 1991/1/1

Y1 - 1991/1/1

N2 - We present an 0(n log2 n) time and 0(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in 0(n log k) time, where k is the combinatorial complexity of the space of transversals and k ≤ 4n. These results find application in: (1) line-fitting between a set of n data ranges where it is desired to obtain the shortest line-of-fit, (2) finding the shortest line segment from which a convex n-vertex polygon is weakly externally visible, and (3) determining the shortest line-of-sight between two edges of a simple n-vertex polygon, for which 0(n) time algorithms are also given. AU the algorithms are based on the solution to a new fundamental geometric optimization problem that is of independent interest and should find application in different contexts as well.

AB - We present an 0(n log2 n) time and 0(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in 0(n log k) time, where k is the combinatorial complexity of the space of transversals and k ≤ 4n. These results find application in: (1) line-fitting between a set of n data ranges where it is desired to obtain the shortest line-of-fit, (2) finding the shortest line segment from which a convex n-vertex polygon is weakly externally visible, and (3) determining the shortest line-of-sight between two edges of a simple n-vertex polygon, for which 0(n) time algorithms are also given. AU the algorithms are based on the solution to a new fundamental geometric optimization problem that is of independent interest and should find application in different contexts as well.

UR - http://www.scopus.com/inward/record.url?scp=69949154918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69949154918&partnerID=8YFLogxK

U2 - 10.1007/3-540-54233-7_171

DO - 10.1007/3-540-54233-7_171

M3 - Conference contribution

AN - SCOPUS:69949154918

SN - 9783540542339

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 649

EP - 660

BT - Automata, Languages and Programming - 18th International Colloquium, Proceedings

PB - Springer-Verlag

ER -